Amazing origami
Copyright (c) 2010 Robert Lang – Soaring Red-Tailed Hawk, opus 601

In our first kirigami paper we outlined the allowed cut-and-fold motifs that were commensurate with an underlying honeycomb lattice. How can we use these motifs to make progress on the “inverse design” problem, where we wish to algorithmically determine the set of cuts and folds that lead to a specified target surface?

In origami this inverse problem has been partially solved. For example, the circle/river packing method first creates the base of the final product and then fills in the remaining folds; alternately, Tomohiro Tachi has proposed the idea of three-dimensional “tucking molecules” as a base unit of construction. There are, nevertheless, certain limits and constraints in these designs. For instance, the fold patterns specified by the circle/river packing algorithms return a pattern whose folded state matches a target surface, but there is no guarantee that any subset of creases can be folded; complex models of this type are typically pre-creased and then folded in a very specific and repetitive sequence. The polygonally-tiled surfaces created by the tucking-molecule method are the product of intricately interlocking crimp folds hidden away beneath the surface. Both of these features reflect the fact that typically complex origami designs do not permit a simple, monotonic, continuous folding motion from the planar state to the desired target state.

The sixon These extremely delicate fold patterns present an obvious challenge to the goal of designing robust self-assembling origami structures, where in many cases extremely fine control of the fold ordering may be required.

small lattice kirigami A hint towards making progress comes from the fact that these fold patterns often “waste” much of the surface to create areas of effective Gaussian curvature, using intricate folds, wedges, and pleats to “remove” material, hiding it beneath the visible surface. Kirigami allows an artful alternative by literally removing that material and then reconnecting the sheet to create dipoles of Gaussian curvature. Our focus here is on the “sixon,” our name for the three-fold degenerate arrangement of 2-4 cuts on the honeycomb lattice shown on the right. A first step is simply to observe that a triangular lattice of these sixons have commensurate fold lines, i.e. a triangular lattice of sixons is an allowed arrangement of kirigami elements. Under the simplest set of choices for mountain and valley folds this lattice leads to alternating up and down triangular plateaus. Purely local changes to these fold patterns (i.e. taking the six folding lines surrounding a given triangle and changing mountain to valley folds and vice versa) allow one to pop one of these triangular plateaus up or down, changing its height by two.

Under this construction the only limitation is that the height of a given triangle must differ by one from every triangle with which it shares an edge. This is a modest restriction (essentially limiting the range of target surfaces to those whose contact angle satisfies an upper bound set by the hexagon size and the triangular lattice spacing). Given a target surface, it is then straightforward to project its height onto a triangulated plane, and fill in the necessary cut-and-fold pattern to approximate it. Since only local moves are required to manipulate the underlying triangular lattice of sixons, the same kirigami sheet can be dynamically reconfigured into a vast array of target surfaces. Shown below is a schematic demonstration of this, selecting the necessary fold pattern to turn a large grid of sixons into either a monkey saddle or Mt. Katahdin. The cut-and-fold patterns for these — or other– target surfaces are available upon request. Algorithmic lattice kirigami